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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Goal: Motivate The Fundamental Equation of QM: Schr\"odinger's Equation}}{1}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{CLUES}}{1}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Quantum Wave Theory}}{1}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{First a detour}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Working with Classical Mechanics, consider the string represented in Figure 1. With no restoring force acting on it, this string is flat. With a restoring force, this string is curved. Let the y-axis be a measure of displacement. From here, let's derive a wave equation.}}{1}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{And Back To The Topic At Hand}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{We don't have a starting equation at the moment. Rather, let's start with the classical story. Suppose we don't know that $\mathaccentV {vec}17E{F}=m\mathaccentV {vec}17E{a}$, but we do know that by studying the wave of a string's motion, we can get $y(x,t)=\qopname  \relax o{sin}(kx-w t)$. What equation is this a solution to?}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Now we should think of $\psi (x,t)=A\qopname  \relax o{sin}(kx-wt)$ where we can find momentum via $P=\hbar k$ and energy via $E=\hbar w$. More broadly, we should see from our fourth clue that:}}{2}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Going back to our third clue that waves superpose, let's consider}}{2}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{We can conclude that if we have a particle}}{2}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{.}}{2}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.3}{What do we now know?}}{2}}
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\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Nowhere in this equation is there $\psi (x,t)^n$ as an acceptable solution. What matters is that as long as $\DOTSI \intop \ilimits@ \limits _{-\infty }^{\infty }|\psi (x,t)|^2\tmspace  +\thinmuskip {.1667em}dx \not =\infty $, $\psi (x,t)$ can be normalized and serve to be a description of the particle in question. What we need to do next is generalize the free particle Schr\"odinger equation (one with no potential V(x)) to a particle that is being acted on by a force.}}{3}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{CLUE 5}}{3}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{What We Know So Far}}{3}}
